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 A024786 Number of 2's in all partitions of n. 39

%I

%S 0,1,1,3,4,8,11,19,26,41,56,83,112,160,213,295,389,526,686,911,1176,

%T 1538,1968,2540,3223,4115,5181,6551,8191,10269,12756,15873,19598,

%U 24222,29741,36532,44624,54509,66261,80524,97446,117862,142029,171036,205290,246211

%N Number of 2's in all partitions of n.

%C Also number of partitions of n-1 with a distinguished part different from all the others. [Comment corrected by _Emeric Deutsch_, Aug 13 2008]

%C In general the number of times that j appears in the partitions of n equals Sum_{k<n, k = n (mod j)} P(k). In particular this gives a formula for a(n), A024787, ..., A024794, for j = 2,...,10; it generalizes the formula given for A000070 for j=1. - Jose Luis Arregui (arregui(AT)posta.unizar.es), Apr 05 2002

%C Equals row sums of triangle A173238. [_Gary W. Adamson_, Feb 13 2010]

%C The sums of two successive terms give A000070. - _Omar E. Pol_, Jul 12 2012

%C a(n) is also the difference between the sum of second largest and the sum of third largest elements in all partitions of n. More generally, the number of occurrences of k in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+1)st largest elements in all partitions of n. And more generally, the sum of the number of occurrences of k, k+1, k+2..k+m in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+m+1)st largest elements in all partitions of n. - _Omar E. Pol_, Oct 25 2012

%C Number of singletons in all partitions of n-1. A singleton in a partition is a part that occurs exactly once. Example: a(5) = 4 because in the partitions of 4, namely [1,1,1,1], [1,1,2'], [2,2], [1',3'], [4'] we have 4 singletons (marked by '). - _Emeric Deutsch_, Sep 12 2016.

%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 184.

%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A024786/b024786.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)

%H Manosij Ghosh Dastidar and Sourav Sen Gupta, <a href="http://arxiv.org/abs/1111.0094">Generalization of a few results in Integer Partitions</a>, arXiv preprint arXiv:1111.0094 [cs.DM], 2011.

%H E. Deutsch et al., <a href="https://www.mat.uniroma2.it/~tauraso/AMM/AMM11237.pdf">Problem 11237</a>, Amer. Math. Monthly, 115 (No. 7, 2008), 666-667. [From _Emeric Deutsch_, Aug 13 2008]

%F a(n) = Sum_{k=1..floor(n/2)} A000041(n-2k). - _Christian G. Bower_, Jun 22 2000

%F a(n) = Sum_{k<n, k = n (mod 2)} P(k), P(k) = number of partitions of k as in A000041, P(0) = 1. - Jose Luis Arregui (arregui(AT)posta.unizar.es), Apr 05 2002

%F G.f.: x^2/((1-x)*(1-x^2)^2))*Product_{j>=3} 1/(1-x^j) from Riordan reference second term, last eq.

%F a(n) = A006128(n-1) - A194452(n-1). - _Omar E. Pol_, Nov 20 2011

%F a(n) = A181187(n,2) - A181187(n,3). - _Omar E. Pol_, Oct 25 2012

%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2) * Pi * sqrt(n)) * (1 - 25*Pi/(24*sqrt(6*n)) + (25/48 + 433*Pi^2/6912)/n). - _Vaclav Kotesovec_, Mar 07 2016, extended Nov 05 2016

%F a(n) = Sum_{k} k * A116595(n-1,k). - _Emeric Deutsch_, Sep 12 2016

%e Contribution from _Omar E. Pol_, Oct 25 2012 (Start):

%e For n = 7 we have:

%e --------------------------------------

%e . Number

%e Partitions of 7 of 2's

%e --------------------------------------

%e 7 .............................. 0

%e 4 + 3 .......................... 0

%e 5 + 2 .......................... 1

%e 3 + 2 + 2 ...................... 2

%e 6 + 1 .......................... 0

%e 3 + 3 + 1 ...................... 0

%e 4 + 2 + 1 ...................... 1

%e 2 + 2 + 2 + 1 .................. 3

%e 5 + 1 + 1 ...................... 0

%e 3 + 2 + 1 + 1 .................. 1

%e 4 + 1 + 1 + 1 .................. 0

%e 2 + 2 + 1 + 1 + 1 .............. 2

%e 3 + 1 + 1 + 1 + 1 .............. 0

%e 2 + 1 + 1 + 1 + 1 + 1 .......... 1

%e 1 + 1 + 1 + 1 + 1 + 1 + 1 ...... 0

%e ------------------------------------

%e . 24 - 13 = 11

%e .

%e The difference between the sum of the second column and the sum of the third column of the set of partitions of 7 is 24 - 13 = 11 and equals the number of 2's in all partitions of 7, so a(7) = 11.

%e (End)

%p b:= proc(n, i) option remember; local f, g;

%p if n=0 or i=1 then [1, 0]

%p else f:= b(n, i-1); g:= `if`(i>n, [0\$2], b(n-i, i));

%p [f[1]+g[1], f[2]+g[2]+`if`(i=2, g[1], 0)]

%p fi

%p end:

%p a:= n-> b(n, n)[2]:

%p seq(a(n), n=1..50); # _Alois P. Heinz_, May 18 2012

%t Table[ Count[ Flatten[ IntegerPartitions[n]], 2], {n, 1, 50} ]

%t b[n_, i_] := b[n, i] = Module[{f, g}, If[n==0 || i==1, {1, 0}, f = b[n, i - 1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + If[i == 2, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Sep 22 2015, after _Alois P. Heinz_ *)

%t Join[{0}, (1/((1 - x^2) QPochhammer[x]) + O[x]^50)[[3]]] (* _Vladimir Reshetnikov_, Nov 22 2016 *)

%t Table[Sum[(1 + (-1)^k)/2 * PartitionsP[n-k], {k, 2, n}], {n, 1, 50}] (* _Vaclav Kotesovec_, Aug 27 2017 *)

%Y Cf. A066633, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794, A173238.

%Y Column 2 of A060244.

%Y First differences of A000097.

%K nonn

%O 1,4

%A _Clark Kimberling_

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Last modified April 20 22:22 EDT 2019. Contains 322310 sequences. (Running on oeis4.)